Partonic quasi-distributions of the pion in chiral quark ...jacobi.fis.ucm.es/REF2017/slides/arriola.pdf · Enrique Ruiz Arriola and Wojciech Broniowski ... Pion quark DF, QM vs

Partonic quasi-distributions of the pionin chiral quark models

Enrique Ruiz Arriola and Wojciech Broniowski

Dep. Fısica Atomica,Molecular y NuclearInstituto Carlos I de Fısica Teorica y Computacional

Universidad de Granada

Resummation, Evolution, Factorization 2017 (REF2017)13-16 November 2017, Madrid

details in Phys.Lett. B773 (2017) 385-390, arXiv:1707.09588

Enrique Ruiz Arriola () Quasi-distributions REF2017 1 / 36

http://www.sciencedirect.com/science/article/pii/S0370269317306779?via%3Dihub

https://arxiv.org/pdf/1707.09588

Outline

Parton distributions – basic properties of hadrons

Soft matrix elements, accessible from low-energy models of QCD

Chiral quark models of the pion

Parton quasi-distributions, designed for Euclidean QCD lattices

Results and predictions for quasi-distributions of the pion from chiral quark models


Introduction

Parton distribution

Q2 = −q2, x = Q2

2p·q , Q2 →∞

Factorization of soft and hard processes,Wilson’s OPE

〈J(q)J(−q)〉=∑i

Ci(Q2;µ)〈Oi(µ)〉

Twist expansion → F (x,Q) = F0(x, α(Q)) + F2(x,α(Q))

Q2 + . . .

Bj limit → light-cone momentum isconstrained: k+ ≡ k0 + k3 = xP+

x ∈ [0, 1]


Parton distribution

Q2 = −q2, x = Q2

2p·q , Q2 →∞

Factorization of soft and hard processes,Wilson’s OPE

〈J(q)J(−q)〉=∑i

Ci(Q2;µ)〈Oi(µ)〉

Twist expansion → F (x,Q) = F0(x, α(Q)) + F2(x,α(Q))

Q2 + . . .

Bj limit → light-cone momentum isconstrained: k+ ≡ k0 + k3 = xP+

x ∈ [0, 1]


Distribution amplitude (DA) of the pion

Enters various measures of exclusive processes,e.g., pion-photon transition form factor


Field-theoretic definition(here for quarks in the pion, leading twist)

Parton Distribution Function (DF):

q(x) =

∫dz−

4πeixP

+z− 〈P |ψ(0)γ+ U [0, z]ψ(z)|P 〉∣∣z+=0,z⊥=0

Parton Distribution Amplitude (DA):

φ(x) =i

Fπ

∫dz−

2πei(x−1)P+z− 〈P |ψ(0)γ+γ5 U [0, z]ψ(z)|vac〉

∣∣z+=0,z⊥=0

(isospin suppressed)

P - pion momentum, v± ≡ v0 ± v3 - light-cone basis

U [z1, z2] = exp(−igs

∫ z2z1dξλaA+

a (ξ))

- Wilson’s gauge link

x - fraction of the light-cone mom. P+ carried by the quark, x ∈ [0, 1]


Remarks

Only indirect experimental information for the pion distributions:

DF from Drell-Yan in E615, DA from dijets in E791 and from exclusive processesinvolving pions

Impossibility to implement PDF or PDA on the euclidean lattices, only lowestmoments can be obtained

However, there exist (largely forgotten) simulations on transverse lattices –discussed later


Quasi-distributions

Parton quasi-distributions (quarks in the pion)

[Ji 2013]

Parton Quasi-Distribution Function (QDF):

V (y;P3) =

∫dz3

4πeiyP

3z3 〈P |ψ(0)γ3 U [0, z]ψ(z)|P 〉∣∣z0=0,z⊥=0

Parton Quasi-Distribution Amplitude (PDA):

φ(y;P3) =i

Fπ

∫dz3

2πei(y−1)P 3z3 〈P |ψ(0)γ+γ5 U [0, z]ψ(z)|vac〉

∣∣z0=0,z⊥=0

y - fraction of pion’s P3 carried by the quarkAnalogy to DF and DA, but y is not constrainedBasic property:

limP3→∞

V (x;P3) = q(x), limP3→∞

φ(x;P3) = φ(x)


QDF and QDA in the momentum representation

Constrained longitudinal momenta, but y ∈ (−∞,∞)(partons can move “backwards”)


Chiral quark models

Chiral quark models

χSB breaking → massive quarks

Point-like interactions

Soft matrix elements with pions (andphotons, W , Z)

One-quark loop, regularization:

1) Pauli-Villars (PV)

2) Spectral Quark Model

(SQM) - implements VMD

Quantities evaluated at the quark model scale(where constituent quarks are the only degrees of freedom)


Chiral quark models

χSB breaking → massive quarks

Point-like interactions

Soft matrix elements with pions (andphotons, W , Z)

One-quark loop, regularization:

1) Pauli-Villars (PV)

2) Spectral Quark Model

(SQM) - implements VMD

Need for evolutionGluon dressing, renorm-group improved


Chiral quark models

Chiral quark models implement chiral symmetry: Pion is a Goldstone boson in thechiral limit

Realization of the large Nc limit

Fully covariant relativistic: Rest frame, light cone, euclidean

Chiral Perturbation Theory: Gasser-Leutwyler coeffs Li ∼ Nc/(4π)2

Chiral anomaly: Wess-Zumino term π0 → 2γ, γ → π0π+π−, etc.

Electromagnetic, Transition, Gravitational Form factors

Parton distribution amplitudes (ϕπ(x) = 1)

Generalized Parton Distributions (dπ(x) = uπ(x) = 1)

Polyakov cooling in chiral phase transitions 〈P 〉 ∼ e−M/T

Baryons as Solitons with dynamical topology (SU(3), ...)


Chiral quark models

Axial current

Jµ,aA (x) =1

2q(x)γµγ5τaq(x) ∂µJµ,aA (x) = −fπm2

ππa (PCAC)

Chiral Ward Identity for irreducible vertex (chiral limit, mπ → 0)

S(p+ q)−1γ51

2τa + γ5

1

2τaS(p)−1 = qµΓµ,aA (p+ q, p)

If S(p) = 1/(/p−M) then a solution

Γµ,aA (p+ q, p) =τa

2γ5

[γµ − qµ

q22M

fπ

]q2 = 0 iff M 6= 0

Pion quark coupling constant (Goldberger-Treiman relation)

gπqq = M/fπ


Wave functions in chiral quark models

Pion Bethe-Salpeter wave function

χaq (p) =i

/p+ q/−M −mgπqqγ5τai

/p−M −m

i g 5 t a

q = 0 , q 0 = m p

i g 5 t aG a

q = 0 , q 0 = m p

p f i x e d

G a

x = 0 , t = 0

x = r , t = 0

( a ) ( b )

ET wave functions

Ψa(r) = −∫

d3p

(2π)3ei~p·~r

∫dp0

(2π)Tr[Γaχq(p)].

LC wave functions

Ψa(~b⊥, x) = −∫

d2p⊥(2π)3

ei~p⊥·~b⊥+ixmπp

+∫

dp−

(2π)Tr[Γaχq(p)].


Equal Time Wave functions in the NJL model

Analytic results for mπ = 0

ΨP (r) = 2ΨT (r) =gπqqNc2π2r

(2MK1(Mr) + regulators) ,

ΨA(r) =gπqqMNc

4π2(2K0(Mr) + regulators)

where K0 and K1 are the modified Bessel functions.

Asymptotic behavior at r →∞:

ΨP (r) ∼ ΨT (r) ∼ e−Mr

r3/2, ΨA(r) ∼ e−Mr

r1/2.

Longer tail in the A channel than in the P and T channels.


Light Cone Wave functions in the NJL model

Analytic results for mπ = 0

ΨP (b, x) = 2ΨT (b, x) =gπqqNc2π2b

(2MK1(Mb) + regulators) ,

ΨA(b, x) =gπqqMNc

4π2(2K0(Mb) + regulators)

These functions are x independent !!

Model Relation between ET and LC

ΨA(b, x) = ΨA(r)|r=b

Pion Distribution Amplitude (flat)

ϕπ(x) = ΨA(0⊥, x) = 1


Form factors (SQM)Spectral representation

S(p) =

∫C

dwρ(w)

/p− w ≡Z(p)

/p− Σ(p)

Vector meson dominance for em form factor

Fem(Q2) = − Nc4π2f2

π

∫dωρ(ω)ω2

∫ 1

0

dx log[ω2 + x(1− x)t

]≡

m2ρ

Q2 +m2ρ

Solution

→ ρV (ω) =1

2πi

1

ω

1

(1− 4ω2/m2ρ)5/2

f2π = Nc

m2ρ

24π2= (87MeV)2

TMD (mπ = 0)

q(x, kT ) =3m3

ρθ(x)θ(1− x)

16π(k2T +m2ρ/4)5/2

→ 〈k2T 〉 =m2ρ

2


Scale and evolution

QM provide non-perturbative result at a low scale Q0

F (x,Q0)|model = F (x,Q0)|QCD , Q0 − the matching scale

Determination of Q0 via momentum fraction: quarks carry 100% of momentum at Q0.One adjusts Q0 in such a way that when evolved to Q = 2 GeV, the quarks carry theexperimental value of 47%

LO DGLAP evolution:

Q0 = 313+20−10 MeV

[Davidson, Arriola 1995]:

q(x;Q0) = 1


Older results from chiral quark models w/ evolution

Pion quark DF, QM vs. E615

LO DGLAP evolution to the scale Q2 = (4 GeV)2:

points: Fermilab E615, Drell-Yan

line: QM evolved to Q = 4 GeV


Pion quark DF, QM vs. transverse lattice

points: transverse lattice[Dalley, van de Sande 2003]

yellow: QM evolved to 0.35 GeV

pink: QM evolved to 0.5 GeV

dashed: GRS param. at 0.5 GeV


Pion DA, QM vs. E791

mπ=0

χQM (Q=Q0)

χQM (Q=2GeV)

8/π x (1 - x)

6x(1-x)

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

x

ϕ(x)

points: E791 data from dijet production in π +A

solid line: QM at Q = 2 GeV

dashed line: asymptotic form 6x(1− x) at Q→∞


Pion DA, QM vs. transverse lattice

■

■

■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■

■

■

mod. μ=0.5GeV

■ tr. lat.

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

x

ϕ(x,μ)

points: transverse lattice data [Dalley, van de Sande 2003]line: QM at Q = 0.5 GeV+

GPD [WB, ERA, Golec-Biernat 2008], TDA [WB, ERA 2007], equal-time (ET) pionwave functions [WB, ERA, S. Prelovsek, L. Santelj 2009] . . .


NEW: Quasi-distributions from QM

Analytic formulas (in the chiral limit)

SQM (at the QM scale):

φ(y, Pz) = V (y, Pz) =1

π

[2mρPzy

m2ρ + 4Pz

2y2+ arctg

(2Pzy

mρ

)]+ (y → 1− y)

(similar simplicity for PV NJL)

Satisfy the proper normalization

∫ ∞−∞

dy φ(y, Pz) =

∫ ∞−∞

dy V (y, Pz) = 1,

∫ ∞−∞

dy 2yV (y, Pz) = 1

and the limit

limPz→∞

φ(y, Pz) = limPz→∞

V (y, Pz) = θ[y(1− y)] = φ(x) = q(x), (y = x)


QDA and QDF from chiral quark models at QM scale

(a)NJL Pz

0.5

1

10

∞

-0.5 0.0 0.5 1.0 1.5

0.0

0.2

0.4

0.6

0.8

1.0

y

ϕ˜(y,Pz)

(b)NJL

-0.5 0.0 0.5 1.0 1.5

0.0

0.5

1.0

1.5

2.0

y2yV(y,Pz)

(a) Quark QDA of the pion in NJL (mπ = 0) at various Pz, plotted vs. y

(b) The same, but for the quark QDF multiplied with 2y


Comparison to Eucidean lattice

○○○○○○○○○○○○○○○○

○

○

○

○

○

○

○

○

○○○

○

○

○

○

○

○

○

○○○○○○○○○○○○○○○○○□□□□□□□□□□□□

□□□□□□□□□□□□□□□

□□□□□□□□□□□□□□□□□□□□□□□□

(a)

NJL

mπ=310MeV

LaMET

Pz=1.3GeV ○

□Pz=0.9GeV

-2 -1 0 1 2 3

0.0

0.2

0.4

0.6

0.8

1.0

1.2

y

ϕ˜(y,Pz)

○○○○○○○○○○○○○○○○

○

○

○

○

○

○

○

○

○○○

○

○

○

○

○

○

○

○○○○○○○○○○○○○○○○○□□□□□□□□□□□□

□□□□□□□□

□

□□□□□□

□□□

□

□□□□□□□□□□□□□□□□□□□□

(b)

SQM

mπ=310MeV

-2 -1 0 1 2 3

0.0

0.2

0.4

0.6

0.8

1.0

1.2

yϕ˜(y,Pz)

Quark QDA of the pion in NJL (a) and SQM (b) (mπ = 310 MeV, Pz = 0.9 and1.3 GeV), plotted vs. y and compared to the lattice at Q = 2 GeV (LaMET [Zhang etal. 2017])


Evolution of QDF

Relation kT -unintegrated quantities (TMA, TMD)

Radyushkin’s formula [2016] – follows just from Lorentz invariance

φ(y, Pz) =

∫ ∞−∞

dk1

∫ 1

0

dxPzTMA(x, k21 + (x− y)2P 2z ).

QDA can be obtained from TMA via a double integration!

Analogously

V (y, Pz) =

∫ ∞−∞

dk1

∫ 1

0

dxPzTMD(x, k21 + (x− y)2P 2z ).


Evolution of unintegrated DFUDF or TMD

Kwiecinski’s method [2003], based on one-loop CCFMDGLAP-like evolution, diagonal in b-space conjugate to kTFor the non-singlet case:

Q2 ∂f(x, b,Q)

∂Q2=

αs(Q2)

2π

∫ 1

0

dz Pqq(z)

×[Θ(z − x) J0[(1− z)Qb] f

(xz, b,Q

)− f(x, b,Q)

]


Results of evolution of pion QDF in Q at fixed Pz

Pz=1GeV

Q2=Q02

Q2=4GeV2

Q2=100GeV2

-0.5 0.0 0.5 1.0 1.50.0

0.2

0.4

0.6

0.8

1.0

y

V(y)

Pz=2GeV

Q2=Q02

Q2=4GeV2

Q2=100GeV2

-0.5 0.0 0.5 1.0 1.50.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

y

V(y)

Strength moved to lower y as Q increases


Changing Pz at fixed Q

Q2=4GeV2Pz=1GeV

Pz=2GeV

Pz=4GeV

Pz=∞

-1.0 -0.5 0.0 0.5 1.0 1.5

-0.1

0.0

0.1

0.2

0.3

0.4

y

yV(y)

Pz →∞ limit achieved fastest at y ∼ 0.6− 0.9


Evolution of QPDA

mπ=0

Pz=0.9GeV

SQM (Q=Q0)

SQM (Q=2GeV)

8/π x (1 - x)

6x(1-x)

-3 -2 -1 0 1 2

0.0

0.2

0.4

0.6

0.8

1.0

y

ϕ˜(y)

mπ=0

Pz=1.3GeV

SQM (Q=Q0)

SQM (Q=2GeV)

8/π x (1 - x)

6x(1-x)

-3 -2 -1 0 1 2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

y

ϕ˜(y)

Quark QDA of the pion in NJL (a) and SQM (b) (mπ = 310 MeV, Pz = 0.9 and1.3 GeV), plotted vs. y and compared to the lattice at Q = 2 GeV (LaMET [Zhang etal. 2017])


Conclusions

Conclusions

Model evaluation of quasi-distributions of the pion (at the QM scale)

Very simple analytic results, all consistency conditions met, illustration and checkof definitions and methods

Results at finite Pz are interesting per se (Radyushkin’s relation, relation toIoffe-time distributions, ET wave functions), can be (favorably) compared to QDAfrom Euclidean lattice QCD

For QDF of the pion predictions made for various Q (Kwiecinski’s evolution) andPz

Pz ∼ 1 GeV, accessible presently on the lattice, may not be sufficiently close toPz →∞ limit

Convergence fastest for intermediate y, suggesting the domain where lattice maywork best

Recent activity also on related objects: pseudo-distributions, Ioffe-timedistributions... Lattice efforts for both N and π


Documents

Partonic quasi-distributions of the pion in chiral quark ...jacobi.fis.ucm.es/REF2017/slides/arriola.pdf · Enrique Ruiz Arriola and Wojciech Broniowski ... Pion quark DF, QM vs